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July  2020, 40(7): 4231-4258. doi: 10.3934/dcds.2020179

Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

1. 

Laboratory Mathématiques et Informatique pour la Complexité et les Systèmes, CentraleSupélec, Univérsité Paris-Saclay, Campus de Gif-sur-Yvette, Plateau de Moulon, 3 rue Joliot Curie, 91190 Gif-sur-Yvette, France

2. 

Institute of Biological, Environmental and Rural Sciences, Aberystwyth University, Penglais Campus, Aberystwyth, Ceredigion, Wales, SY23 3DA, United Kingdom

* Corresponding author: Anna Rozanova-Pierrat and Vladimir Khodygo

Received  May 2019 Revised  January 2020 Published  April 2020

We relate together different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the $ L^2 $ norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.

Citation: Adrien Dekkers, Anna Rozanova-Pierrat, Vladimir Khodygo. Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4231-4258. doi: 10.3934/dcds.2020179
References:
[1]

S. I. AanonsenT. BarkveJ. N. Tjøtta and S. Tjøtta, Distortion and harmonic generation in the nearfield of a finite amplitude sound beam, The Journal of the Acoustical Society of America, 75 (1984), 749-768. 

[2]

R. A. Adams, Sobolev Spaces, Vol. 65, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

S. Alinhac, Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.  doi: 10.1007/BF01231301.

[4]

S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), (2002), Exp. No. Ⅰ, 33 pp.

[5]

N. S. Bakhvalov, Y. M. Zhileǐkin and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams, American Institute of Physics Translation Series, American Institute of Physics, New York, 1987.

[6]

L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, Providence, RI, 1979.

[7]

P. Caine and M. West, A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional cartesian version without use of perturbation expansions, Applied Acoustics, 45 (1995), 155-165. 

[8]

Z. CaoH. YinL. Zhang and L. Zhu, Large time asymptotic behavior of the compressible Navier-Stokes equations in partial space-periodic domains, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1167-1191.  doi: 10.1016/S0252-9602(16)30061-3.

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.

[10]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Vol. 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4$^th$ edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.

[11]

A. Dekkers and A. Rozanova-Pierrat, Cauchy problem for the Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 277-307.  doi: 10.3934/dcds.2019012.

[12]

A. Dekkers and A. Rozanova-Pierrat, Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems, preprint, arXiv(1811.10850).

[13]

R. DenkM. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.

[14]

G. Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl., 138 (1984), 55-104.  doi: 10.1007/BF01762539.

[15]

M. GhisiM. Gobbino and A. Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.  doi: 10.1090/tran/6520.

[16]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[17]

B. Gustafsson and A. Sundström, Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 343-357.  doi: 10.1137/0135030.

[18] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, 1998. 
[19]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.  doi: 10.1007/BF00390346.

[20]

D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354.  doi: 10.1007/s002050050055.

[21]

K. Ito, Smooth global solutions of the two-dimensional Burgers equation, Canad. Appl. Math. Quart., 2 (1994), 283-323. 

[22]

F. John, Nonlinear Wave Equations, Formation of Singularities, Vol. 2, University Lecture Series, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002.

[23]

P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.  doi: 10.1016/j.physleta.2004.03.067.

[24]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.

[25]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., (2011), 763–773.

[26]

B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 34 pp. doi: 10.1142/S0218202512500352.

[27]

B. Kaltenbacher and V. Nikolić, The Jordan-Moore-Gibson-Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.  doi: 10.1142/S0218202519500532.

[28]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part Ⅰ. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.

[29]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; Exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.

[30]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.

[31]

V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Phys. Acoust., 16 (1971), 467-470. 

[32]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.

[33]

M. J. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528. 

[34]

T. LuoC. Xie and Z. Xin, Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.  doi: 10.1016/j.aim.2015.12.027.

[35]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part Ⅱ, Acta Acustica United with Acustica, 83 (1997), 197-222. 

[36]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738.

[37]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[38]

B. E. McDonaldP. Caine and M. West, A tutorial on the nonlinear progressive wave equation (NPE)–part 1, Applied Acoustics, 43 (1994), 159-167. 

[39]

B. E. McDonald and W. A. Kuperman, Time-domain solution of the parabolic equation including nonlinearity, Comput. Math. Appl., 11 (1985), 843-851.  doi: 10.1016/0898-1221(85)90179-8.

[40]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.

[41]

A. Rozanova-Pierrat, Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Math. Models Methods Appl. Sci., 18 (2008), 781-812.  doi: 10.1142/S0218202508002863.

[42]

A. Rozanova-Pierrat, On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media, Commun. Math. Sci., 7 (2009), 679-718.  doi: 10.4310/CMS.2009.v7.n3.a9.

[43]

A. Rozanova-Pierrat, Approximation of a compressible Navier-Stokes system by non-linear acoustical models, Proceedings of the International Conference DAYS on DIFFRACTION, (2015), 270–276.

[44]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.

[45]

T. C. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.  doi: 10.1512/iumj.1991.40.40025.

[46]

T. C. Sideris, The lifespan of 3D compressible flow, Séminaire sur les Équations aux Dérivées Partielles, 5 (1992), 10 pp.

[47]

T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.  doi: 10.1353/ajm.1997.0014.

[48]

M. F. Sukhinin, On the solvability of the nonlinear stationary transport equation, Teoret. Mat. Fiz., 103 (1995), 23-31.  doi: 10.1007/BF02069780.

[49]

J. N. Tjøtta and S. Tjøtta, Nonlinear equations of acoustics, with application to parametric acoustic arrays, The Journal of the Acoustical Society of America, 69 (1981), 1644–1652.

[50]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537. 

[51]

H. Yin and Q. Qiu, The lifespan for 3-D spherically symmetric compressible sEuler equations, Acta Math. Sinica (N.S.), 14 (1998), 527-534.  doi: 10.1007/BF02580410.

show all references

References:
[1]

S. I. AanonsenT. BarkveJ. N. Tjøtta and S. Tjøtta, Distortion and harmonic generation in the nearfield of a finite amplitude sound beam, The Journal of the Acoustical Society of America, 75 (1984), 749-768. 

[2]

R. A. Adams, Sobolev Spaces, Vol. 65, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

S. Alinhac, Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.  doi: 10.1007/BF01231301.

[4]

S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), (2002), Exp. No. Ⅰ, 33 pp.

[5]

N. S. Bakhvalov, Y. M. Zhileǐkin and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams, American Institute of Physics Translation Series, American Institute of Physics, New York, 1987.

[6]

L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, Providence, RI, 1979.

[7]

P. Caine and M. West, A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional cartesian version without use of perturbation expansions, Applied Acoustics, 45 (1995), 155-165. 

[8]

Z. CaoH. YinL. Zhang and L. Zhu, Large time asymptotic behavior of the compressible Navier-Stokes equations in partial space-periodic domains, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1167-1191.  doi: 10.1016/S0252-9602(16)30061-3.

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.

[10]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Vol. 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4$^th$ edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.

[11]

A. Dekkers and A. Rozanova-Pierrat, Cauchy problem for the Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 277-307.  doi: 10.3934/dcds.2019012.

[12]

A. Dekkers and A. Rozanova-Pierrat, Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems, preprint, arXiv(1811.10850).

[13]

R. DenkM. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.

[14]

G. Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl., 138 (1984), 55-104.  doi: 10.1007/BF01762539.

[15]

M. GhisiM. Gobbino and A. Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.  doi: 10.1090/tran/6520.

[16]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[17]

B. Gustafsson and A. Sundström, Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 343-357.  doi: 10.1137/0135030.

[18] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, 1998. 
[19]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.  doi: 10.1007/BF00390346.

[20]

D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354.  doi: 10.1007/s002050050055.

[21]

K. Ito, Smooth global solutions of the two-dimensional Burgers equation, Canad. Appl. Math. Quart., 2 (1994), 283-323. 

[22]

F. John, Nonlinear Wave Equations, Formation of Singularities, Vol. 2, University Lecture Series, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002.

[23]

P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.  doi: 10.1016/j.physleta.2004.03.067.

[24]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.

[25]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., (2011), 763–773.

[26]

B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 34 pp. doi: 10.1142/S0218202512500352.

[27]

B. Kaltenbacher and V. Nikolić, The Jordan-Moore-Gibson-Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.  doi: 10.1142/S0218202519500532.

[28]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part Ⅰ. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.

[29]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; Exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.

[30]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.

[31]

V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Phys. Acoust., 16 (1971), 467-470. 

[32]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.

[33]

M. J. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528. 

[34]

T. LuoC. Xie and Z. Xin, Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.  doi: 10.1016/j.aim.2015.12.027.

[35]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part Ⅱ, Acta Acustica United with Acustica, 83 (1997), 197-222. 

[36]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738.

[37]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[38]

B. E. McDonaldP. Caine and M. West, A tutorial on the nonlinear progressive wave equation (NPE)–part 1, Applied Acoustics, 43 (1994), 159-167. 

[39]

B. E. McDonald and W. A. Kuperman, Time-domain solution of the parabolic equation including nonlinearity, Comput. Math. Appl., 11 (1985), 843-851.  doi: 10.1016/0898-1221(85)90179-8.

[40]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.

[41]

A. Rozanova-Pierrat, Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Math. Models Methods Appl. Sci., 18 (2008), 781-812.  doi: 10.1142/S0218202508002863.

[42]

A. Rozanova-Pierrat, On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media, Commun. Math. Sci., 7 (2009), 679-718.  doi: 10.4310/CMS.2009.v7.n3.a9.

[43]

A. Rozanova-Pierrat, Approximation of a compressible Navier-Stokes system by non-linear acoustical models, Proceedings of the International Conference DAYS on DIFFRACTION, (2015), 270–276.

[44]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.

[45]

T. C. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.  doi: 10.1512/iumj.1991.40.40025.

[46]

T. C. Sideris, The lifespan of 3D compressible flow, Séminaire sur les Équations aux Dérivées Partielles, 5 (1992), 10 pp.

[47]

T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.  doi: 10.1353/ajm.1997.0014.

[48]

M. F. Sukhinin, On the solvability of the nonlinear stationary transport equation, Teoret. Mat. Fiz., 103 (1995), 23-31.  doi: 10.1007/BF02069780.

[49]

J. N. Tjøtta and S. Tjøtta, Nonlinear equations of acoustics, with application to parametric acoustic arrays, The Journal of the Acoustical Society of America, 69 (1981), 1644–1652.

[50]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537. 

[51]

H. Yin and Q. Qiu, The lifespan for 3-D spherically symmetric compressible sEuler equations, Acta Math. Sinica (N.S.), 14 (1998), 527-534.  doi: 10.1007/BF02580410.

Table 1.  Approximation results for models derived from the Kuznetsov equation
KZK NPE Westervelt
periodic boundary condition problem initial boundary value problem viscous and inviscid case viscous case inviscid case
Theorem Theorem 2.1 Theorem 2.2 Theorem 3.1 Theorem 4.2
Derivation paraxial approximation
$ u=\Phi(t-\frac{x_1}{c},\varepsilon x_1,\sqrt{\varepsilon} \textbf{x}') $
paraxial approximation
$ u=\Psi(\varepsilon t,x_1-ct,\sqrt{\varepsilon} \textbf{x}') $
$ \Pi=u+\frac{1}{c^2}\varepsilon u \partial_t u $
Approximation domain the half space
$ \{x_1>0, x'\in {\mathbb{R}}^{n-1}\} $
$ \mathbb{T}_{x_1}\times\mathbb{R}^2 $ $ \mathbb{R}^n $
Approximation order $ O(\varepsilon) $ $ O(\varepsilon) $ $ O(\varepsilon^2) $
Estimation $ \Vert I-I_{aprox}\Vert_{L^2(\mathbb{T}_t\times\mathbb{R}^{n-1})}\leq \varepsilon $
$ z\leq K $
$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon. $
$ t<\frac{T}{\varepsilon} $
$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon $
$ t<\frac{T}{\varepsilon} $
$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon $
$ t<\frac{T}{\varepsilon} $
Initial data regularity $ I_0\in H^{s+\frac{3}{2}}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ s> \max(\frac{n}{2},2) $
$ I_0\in H^{s}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ \left[\frac{s}{2}\right]>\frac{n}{2}+2 $
$ \xi _0\in H^{s+2}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $
for $ s>\frac{n}{2}+1 $
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+3}(\mathbb{R}^3) $
for $ s>\frac{n}{2} $
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+2}(\mathbb{R}^3) $
for $ s>\frac{n}{2} $
Data regularity for remainder boundness $ I_0\in H^{s+\frac{3}{2}}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ s> \max(\frac{n}{2},2) $
$ I_0\in H^{6}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ n= 2,3 $,
$ I_0\in H^{s}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ \left[\frac{s}{2}\right]>\frac{n}{2}+1 $, $ n\geq 4 $
$ \xi _0\in H^{4}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $
for $ n=2,3 $.
$ \xi _0\in H^{s}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $
for $ s>\frac{n}{2}+2 $, $ n\geq4 $.
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+3}(\mathbb{R}^n) $
for $ s>\frac{n}{2} $
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+2}(\mathbb{R}^n) $
for $ s>\frac{n}{2} $
KZK NPE Westervelt
periodic boundary condition problem initial boundary value problem viscous and inviscid case viscous case inviscid case
Theorem Theorem 2.1 Theorem 2.2 Theorem 3.1 Theorem 4.2
Derivation paraxial approximation
$ u=\Phi(t-\frac{x_1}{c},\varepsilon x_1,\sqrt{\varepsilon} \textbf{x}') $
paraxial approximation
$ u=\Psi(\varepsilon t,x_1-ct,\sqrt{\varepsilon} \textbf{x}') $
$ \Pi=u+\frac{1}{c^2}\varepsilon u \partial_t u $
Approximation domain the half space
$ \{x_1>0, x'\in {\mathbb{R}}^{n-1}\} $
$ \mathbb{T}_{x_1}\times\mathbb{R}^2 $ $ \mathbb{R}^n $
Approximation order $ O(\varepsilon) $ $ O(\varepsilon) $ $ O(\varepsilon^2) $
Estimation $ \Vert I-I_{aprox}\Vert_{L^2(\mathbb{T}_t\times\mathbb{R}^{n-1})}\leq \varepsilon $
$ z\leq K $
$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon. $
$ t<\frac{T}{\varepsilon} $
$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon $
$ t<\frac{T}{\varepsilon} $
$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon $
$ t<\frac{T}{\varepsilon} $
Initial data regularity $ I_0\in H^{s+\frac{3}{2}}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ s> \max(\frac{n}{2},2) $
$ I_0\in H^{s}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ \left[\frac{s}{2}\right]>\frac{n}{2}+2 $
$ \xi _0\in H^{s+2}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $
for $ s>\frac{n}{2}+1 $
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+3}(\mathbb{R}^3) $
for $ s>\frac{n}{2} $
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+2}(\mathbb{R}^3) $
for $ s>\frac{n}{2} $
Data regularity for remainder boundness $ I_0\in H^{s+\frac{3}{2}}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ s> \max(\frac{n}{2},2) $
$ I_0\in H^{6}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ n= 2,3 $,
$ I_0\in H^{s}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $
for $ \left[\frac{s}{2}\right]>\frac{n}{2}+1 $, $ n\geq 4 $
$ \xi _0\in H^{4}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $
for $ n=2,3 $.
$ \xi _0\in H^{s}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $
for $ s>\frac{n}{2}+2 $, $ n\geq4 $.
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+3}(\mathbb{R}^n) $
for $ s>\frac{n}{2} $
$ u_0\in H^{s+3}(\mathbb{R}^n) $
$ u_1\in H^{s+2}(\mathbb{R}^n) $
for $ s>\frac{n}{2} $
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